Macromolecules 1995,28, 4740-4741
4740
Determination of Free-Volume Parameters for Solvent Self-Diffusion in Polymer-Solvent Systems J. S. Vrentas"and C. M. Vrentas Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 Received January 31, 1995 Revised Manuscript Received April 17, 1995
Introduction In two recent papers,1,2a free-volume theory of transport was developed fer predicting and correlating solvent self-diffusion coefficients for both rubbery and glassy polymer-solvent systems. For a rubbery polymer-solvent system, it was shown that concentration and temperature variations of the solvent self-diffusion coefficient, D1, can be determined using the following equations:l
D, = Do exp
[
--
[
exp -
wlV1*
+ W,;V2*]
(1)
VFHfY
(7)
V2* = E(0)
(9)
In these equation;, DOis an effectively constant preexponential factor, VI* is the specific hole free volume of component I required for a jump, 01 is the mass fraction of component I, TgI is the glass transition temperature of pure component I, T is the temperature, a n d E * is an energy parameter defined e1sewhere.l Also, VCH is the average hole free volume per gram of mixture, VFH2 is the specific hole free volume of the equilibrium liquid is the fractional hole polymer at any temperature, free volume of the polymer at its glass transition temperature Tg2,y represents an average overlap factor
g2
system polystyrene- toluene polystyrene- toluene polystyrene-ethylbenzene polystyrene-ethylbenzene
for the mixture, and y~ represents the overlap factor for the free volume of pure component I. Furthermore, K I ~ and K2l are free-volume parameters for the solvent, K12 and K22 are free-volume parameters for the polymer, p(0)is the specific volume of component I at 0 K, E ( T g 2 is ) the specific volume of the liquid polymer at Tg2, ac2 is the thermal expansion coefficient for the polymer for the sum of the specific occupied volume and the specific interstitial free volume, and 5 is the ratio of the critical molar volume of the solvent jumping unit to the critical molar volume of the polymer jumping unit. Finally, a2 is the thermal expansion coefficient for the equilibrium liquid polymer. In eqs 3 and 7, an average value of a2 above Tg2is used. In eq 4,a temperaturedependent az(T)should be used if the appropriate data are available. From the above equations, it is evident that the dependence of D1 on w1 and T for rubbery polymersolvent systems can be computed if the following t,K d parameters are available: D o , E*, q ( O ) , YI, KZI - Tgl, K1dy2, KZZ,T o , %Tg2), and adT). Furthermore, it has been shown elsewhere2 that these parameters are sufficient for the calculation of D1 for glassy polymer-solvent systems. All of the above parameters, with the exception of 5 and E*, can be determined using generally available data for the pure components. A parameter evaluation scheme has been presented elsewhere,l and it is noted there that as few as two diffusivity data points can be used to determine 5 and E*. In the procedure proposed previously, 5 and E* can be determined using measured solvent selfdiffusion coefficients at w1 = 0 and at two or more temperatures. Such a procedure can be easily applied to a inverse gas chromatography experiment where infinite dilution diffusion coefficients can be measured over relatively wide temperature ranges. In addition, it is possible to obtain Dl(wl=O) values for determining 5 and E* by extrapolating diffisivity data collected using a high-temperature vapor sorption experiment. However, it is usually difficult to extract Dl(wl=O) values from a low-temperature vapor sorption experiment because the strong concentration dependence of diffusion coeffkients near w1 = 0 effectively precludes any definitive extrapolation to 01 = 0. Furthermore, solvent self-diffusion data, measured using NMR experiments for moderate to high solvent concentrations (typically for 01 > 0.11, cannot be used because extrapolation to w1 = 0 is again not feasible. The purpose of this note is to present an alternative method for determining 5 and E* from solvent selfdiffusion data. This procedure, which utilizes at least two D1 - 01 data points at a single temperature, can be used to analyze NMR self-diffusion experiments and both low- and high-temperature vapor sorption experiments. The proposed procedure thus permits the utilization of a wider variety of data sets in the evaluation of 5 and E*. This procedure is presented and illustrated in the next section.
E(O),
Table 1. Evaluation of Parameters temD ("C) expt evaluation method 160, 170, 178 vapor sorption temp dependence of D1 ( w 1 = 0) concn dependence a t single T NMR 115 temp dependence of D1 (mi= 0) vapor sorption 130, 140 concn dependence a t single T NMR 100
0024-929719512228-4740$09.00/0
5
E* [kcal/(g mol)]
0.575 0.540
0 0.08 0.61 0.42
0.590
0.583
0 1995 American Chemical Society
Macromolecules, Vol. 28, No. 13, 1995
Notes 4741
Evaluation of Parameters The concentration dependence of DI at a single temperature can be measured either by using an NMR experiment to directly measure the solvent self-diffusion coefficient D1 or a vapor sorption experiment to measure the concentration dependence of the mutual diffusion coefficient D. In the latter case, it is possible to then convert3 D to D1; it is preferable to do this conversion at relatively low values of 01. The D1 versus 01 data at a single temperature can then be used to determine ( and E* by writing eq 1 in the following form for temperatures above Tg2:
Y = @ + -E*
RT
w2v2*
X=-
VFH~
(12)
Both Y and X can be calculated using D1 versus 01 data, eqs 2, 3, and 6-9, and the following previously determined parameters: q ( O ) , &, Klllyl, K Z I- Tgl, KZZ,K d y 2 , $(Tg2), Tg2, and a2. Self-diffision data above Tg2 are used to avoid having to estimate az(T) below Tg2. Clearly, these data can be put in a straightline form with a slope equal to ( and an intercept equal to E*IRT. The proposed method is illustrated using self-diffusion data for the polystyrene-toluene and polystyreneethylbenzene systems. The free-volume parameters ( and E* were previously determinedl for these systems by using Dl(ol=O) data obtained over a small temperature range from a vapor sorption experiment and by using the previous parameter evaluation procedure. The results are summarized in Table 1. The ( and E* parameters for these two systems can also be determined by using the new method and D1 versus 01 data a t a single temperature. For the polystyrene-toluene system, three D1 data point^^,^ at 115 "C were used with eq 10, and for the polystyrene-ethylbenzene system,
e(O),
two D1 data points6 at 100 'C were used in the parameter evaluation. The new procedure based on eq 10 was used to evaluate ( and E* values for each system, and these results are presented in Table 1. There is reasonably good agreement between the parameters evaluated using the previous temperaturedependence method and the present concentrationdependence method, even though both the methods and data sets used are very different. This agreement not only provides some verification of the general applicability of the free-volume theory of diffusion, but it suggests that the proposed method based on the concentration dependence of D1 is a viable alternative to the previous method based on the temperature dependence of D1. Furthermore, the present method can be used to extract ( and E* from NMR data and from both low- and high-temperature vapor sorption data, and data are needed,only at one temperature. In addition, no extrapolation of data is required. Consequently, it is fair to conclude that the proposed method is an attractive alternative which provides increased flexibility in the parameter evaluation scheme. Finally, we note that Waggoner et al.7have suggested a method of determining E only based on using the concentration dependence of D1 and on using the value of D1 for the pure solvent as a reference. This method assumes that the polymer contribution to the free volume is negligible.
Acknowledgment. This work was supported by funds provided by the Dow Chemical Co. References and Notes (1) Vrentas, J. S.; Vrentas, C. M. Macromolecules 1994,27, 4684. (2) Vrentas, J. S.; Vrentas, C. M. Macromolecules 1994,27, 5570. (3) Vrentas, J. S.; Vrentas, C. M. Macromolecules 1993,26, 6129. (4) Pickup, S. Ph.D. Thesis, Drexel University, Philadelphia, PA, 1987. (5) Pickup, S.; Blum, F. D. Macromolecules 1989,22,3961. (6) Zgadzai, 0. E.; Maklakov, A. I. Acta Polym. 1985,36,621. (7) Waggoner, R. A.; Blum, F. D.; MacElroy, J. M. D. Macromolecules 1993,26,6841. MA9501151
